Chevalley-Warning Theorem

Let $\mathbb{F}_{q}$ be the finite field of $q$ elements with characteristic $p$ . Let $f_{i}(x_{1},\ldots,x_{n})$ , $i=1,2,\ldots,r$ , be polynomial of $n$ variables over $\mathbb{F}_{q}$ . If $n>\sum_{i=1}^{r}\deg(f_{i})$ , then the number of solutions over $\mathbb{F}_{q}$ to the system of equations

	$\displaystyle f_{1}(x_{1},x_{2},\ldots,x_{n})$	$\displaystyle=0$
	$\displaystyle f_{2}(x_{1},x_{2},\ldots,x_{n})$	$\displaystyle=0$
	$\displaystyle\vdots$
	$\displaystyle f_{r}(x_{1},x_{2},\ldots,x_{n})$	$\displaystyle=0$

is divisible by $p$ . In particular, if none of the polynomials $f_{1}$ , $f_{2},\ldots,f_{r}$ have constant term, then there are at least $p$ solutions.

Title	Chevalley-Warning Theorem
Canonical name	ChevalleyWarningTheorem
Date of creation	2013-03-22 17:46:52
Last modified on	2013-03-22 17:46:52
Owner	kshum (5987)
Last modified by	kshum (5987)
Numerical id	6
Author	kshum (5987)
Entry type	Theorem
Classification	msc 12E20